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Unit 2: Getting Down to the Basics

2 Introduction

This unit is concerned with numbers, in particular how to write and understand them. We will also look at how you can work with numbers to make it easier to make quick calculations in your head or on paper.

The unit will also focus on strategies you can use when reading and getting used to the language of mathematics. (Yes, math is a language!) So let’s approach learning math as learning a new language—a brand new way to communicate your ideas and thoughts, and to share those with your friends and other people.

Practice Quiz

To check your understanding of place value and rounding of numbers before you start, give the Unit 2 pre quiz a try, then use the feedback to help you plan your study.

The quiz does not check all the topics in the unit, but it should give you some idea of the areas you may need to spend most time on. Remember, it doesn’t matter if you get some or even all of the questions wrong—it just indicates how much time you may need for this unit!

Click here for the pre quiz.

2.0.1 What to Expect in this Unit

This unit should take around eight hours to complete. In this unit you will learn about:

  • The history of numbers.
  • How we put numbers together including decimals.
  • Number lines and how to use them.
  • Rounding numbers and estimating.

As with any language, we will start with the basics. In Section 1, we will explore the history of numbers, including the use of “placeholders” and decimals. It may surprise you to learn that our number system continues to develop even today.

In Section 2, the main focus will be reading and using a number line. Whether you are practicing writing numbers or participating in a competition to see whose flower grows the tallest, you will continue to develop the skills you need to perform future calculations in this section.

In Section 3, you will study rounding, which will help you give appropriate answers to questions and make good estimates of the correctness of your answer. In the sciences especially, rounding can significantly affect your answer, and that is something you need to be aware of. For example, as a nurse, you wouldn’t want to administer a dangerous amount of medication to a patient over a certain period of time!

2.0.2 Where Do You See Numbers?

Activity symbol Activity: Numbers, Numbers Everywhere

You can see that numbers are everywhere! Can you think of some examples where you see or use numbers?

Hint symbol
Discussion

Look around you! Check out the floor, and look at what’s in your desk, wallet, or purse. Notice signs along the road. You’ll be surprised at how many numbers you’ll find! How many can you see from where you’re sitting?

Solution symbol
Answer

Numbers pop up in all kinds of places, such as in check books, on subway tickets, on billboard signs, on your computer keyboard and cell phone … just about everywhere you look!

2.0.3 Using Numbers

Think about an everyday task such as planning a journey. You must have numbers for your journeys so that you can:

  • Indicate the time and dates for the beginning and end of your journey.
  • Calculate the distance.
  • Estimate the time and fuel you would need.
  • Figure out how much money you would need for food, lodging, and entertainment.

Like it or not, numbers play an essential role in our lives, each and every day! Whether you are looking at buying a new car or trying to figure out how much new carpet will cost, you need numbers.

Videoclip symbolBefore we move on, let’s see a short video on how numbers and math are used every day.

Interactive feature not available in single page view (see it in standard view).

2.0.4 Working Through Unit 2

As you work through the unit, use some of the ideas mentioned in Unit 1 for learning actively. Remember that taking breaks will be important. You will have to determine how long you can study this material before you need to stop and relax. Be sure to take advantage of all of your resources—friends, family, the Internet.

Pay close attention to how math is written in the unit and how you write the corresponding details in your math notebook. Remember that math is a language, so how we write it really does matter.

2.1 History of Numbers

Let’s start this chapter with some history. The numbers we use today, called Hindu-Arabic numbers, are a combination of just ten symbols or digits: 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. These digits were introduced in Europe during the thirteenth century by Leonardo Pisano (also known as Fibonacci), an Italian mathematician. Pisano was educated in North Africa, where he learned and later carried to Italy the now popular Hindu-Arabic numerals.

The Hindu numeral system is a pure place value (or base) system, which is why you need a zero, as we’ll see later.

[ Did you know that a symbol representing “nothing” has been found on tablets dating from 700 bce? It wasn’t until much later that the symbol we know as zero was adopted and used systematically. ] Mathematician Bertrand Russell called the invention of zero “the greatest achievement of the human mind.”

And not everybody needs numbers. It’s said that there’s a tribe in the Trobriand Islands in the South Pacific who only have three numbers: “one,” “two,” and “many.” Works for them …

One of the main reasons the number system developed and continues to develop is so that people can use numbers to solve a wide variety of important problems in trading, building, and navigating, although in this chapter, we will be concentrating on more everyday problems.

2.1.1 Written Numbers Compared

Below is a table that shows how numerals changed over time as they traveled from India to Europe.

Activity sumbol Activity: Brief History of Numbers

[ There may be some ideas, such as negative numbers, that you have not met before. Don’t worry about this now—we’ll return to this topic later. The idea now is to be comfortable with the idea of numbers. ]

Audio icon Listen to the following audio track, which is about the history of numbers. In your math notebook, briefly summarize how the number system developed.

Download this audio clip.Audio player: track02.mp3
Skip transcript

Transcript

Y162 Starting with Maths

Audio track 2

A brief history of numbers

Alice Peasgood

Have you ever wondered where numbers come from and why we write them in the way that we do? It's so easy to take them for granted but in fact it has taken tens of thousands of years for numbers to develop and many different cultures have been involved. Hilary, the course team Chair, is here with me to talk about the history of numbers and how people have thought about numbers through the centuries and developed different ways of representing them in writing. When I first learnt about numbers as a child, the first thing that I did was learn how to count. I mean is that actually what has happened historically as well?

Hilary Holmes

Well counting is certainly a very important part as people do need to record sort of how many animals they have got and so on and you can imagine initially people would probably use their fingers or parts of the body to keep a note of how many different things they had and that has actually occurred all across the world from Greenland to New Guinea to Africa and so on. So yes, counting is a very important first step.

Alice Peasgood

But the thing about counting with parts of the body like your fingers, there is no record of that so what's the first record we have that somebody was actually counting?

Hilary Holmes

Well, the first piece of evidence is really a bone which was found in Central Africa at a place called Ishango, and it's known as the Ishango Bone for that reason and on there, there are marks made in lines that are actually grouped together in particular combinations and historians feel that that is evidence that people were interested in counting that many thousands of years ago.

Alice Peasgood

So what you are saying is that it's not the fact that there are scratches on this bone, it's the fact that they are in groups and it looks like there’s a pattern that means someone must have been thinking about the numbers when they made the scratches. That was twenty thousand years ago—I mean, what happened next? Just making scratches on bone doesn’t take you very far. What’s the next step that we have on record?

Hilary Holmes

Well there is a limit to how many numbers you could record in that way and the next main development actually happened in the Middle East, about twelve thousand years ago, where people decided to use small, clay tokens to represent numbers. So for example a small cone would represent the number one. So if you have four goats you would represent that by the four small cones.

Alice Peasgood

So these were a bit like counters, so they’re small enough to hold in your hand, and if you wanted to trade your goats you could take four of these little clay counters along and trade with those.

Hilary Holmes

And they also had other tokens of different sizes and shapes to represent other numbers. So for example, a large cone would actually represent 60, and if that large cone then had holes in it would represent the number 3600.

Alice Peasgood

How would that work if you had different numbers?

Hilary Holmes

Well then you would have to combine the small and large cones. So for example, if I gave you two large cones and three small cones the two large cones would represent two lots of sixty which is one hundred and twenty and then the three small cones would be another three. So altogether that would represent one hundred and twenty three.

Alice Peasgood

Oh right. So instead of having a lot of tallies really all you have to do is have a handful of these clay tokens and actually you can represent some pretty big numbers with them.

Hilary Holmes

That’s right, and that’s why it was important to have different cones to represent different amounts because obviously if you were trying to represent a number like one hundred and twenty three, the last thing you would want really is one hundred and twenty three small cones.

Alice Peasgood

But that’s still not writing anything down is it? It's still just having collections of cones. I mean did they ever write anything down about this?

Hilary Holmes

Well they did but it actually took another five thousand years. What actually happened was when they were doing trading, they would put the cones into jars, to keep them safe as a record as I am sure you can imagine people could lift the cones out or put extra cones in so it wasn’t a particularly secure system. So rather than just putting things into jars they then decided that it would be safer to enclose the tokens in clay and then people couldn’t take them in or out. But there was a problem there because once they had covered them in clay they couldn’t tell what was actually inside. So at that point they decided to make marks on the outside of the clay to show which cones were inside. And from that it was only a small step to decide to record those marks actually on to a clay tablet and that was the beginning of recording numbers and really the beginning of writing.

Alice Peasgood

The thing that puzzles me in all of this is that they are using sixty for their counting. They are going from one to sixty to three thousand six hundred, which is sixty times sixty, but we use tens as our basis so that does seem a bit odd.

Hilary Holmes

But if you think about how we measure time we still use a base sixty system in that we have got sixty seconds in a minute and sixty minutes in an hour. The other reason for using sixty is that you can actually divide a lot of numbers into sixty easily. So it would have made some calculations easier.

Alice Peasgood

Yes because it is quite handy when you are dividing an hour into minutes because you do half an hour, quarter of an hour, twenty minutes, whatever you want to. It divides quite well. That’s covered different sorts of counting but what about if you wanted to talk about parts of something say fractions? What's happened there?

Hilary Holmes

Well there we need to move across to Egypt and see how the Egyptians were using number and they were very interested in building and measurement so they started to develop the use of fractions and we actually know this from an old papyrus called the Rind Papyrus, which is currently in the British Museum which shows some of these fractions and other numbers on it.

Alice Peasgood

But papyrus is a form of paper. Because I was wondering about how these records have actually survived as well?

Hilary Holmes

It is very fragile and there is the Rind Papyrus and also various other documents that historians have been able to decipher. This papyrus explains a lot of how the Egyptians were using number, in particular in the ways of calculating by doubling and halving numbers but also how they used fractions and the fractions that they tended to use were what we call unit fractions, which is just where you have one part of a whole like a quarter, half or one third. And they would use those fractions to build up other fractions.

Alice Peasgood

So if the Egyptians just used fractions which only had one at the top as unit fractions, how did they handle a number like three-quarters?

Hilary Holmes

Well there they would just add unit fractions together to make the fraction they wanted. So for example if you look at three-quarters, you can think of that as a half plus an extra quarter. So they would have expressed three quarters as a half plus a quarter.

Alice Peasgood

Oh I see. So all they do is keep adding together small fractions until they get the large one that they need.

Hilary Holmes

That’s right. And they would have used documents to help them do that.

Alice Peasgood

Where did our modern numbers, one to nine, come from?

Hilary Holmes

About two and half thousand years ago in India they started to use the numbers one to nine but it actually took another thousand years before they used them in the place value system that we have today.

Alice Peasgood

What's the place value system?

Hilary Holmes

That's just where the position of the digit represents its value. So for example if you look at the number twenty-three that means two lots of tens and three units. One of the first places that this place value system was actually recorded was in 458 AD in an Indian book on cosmology which was called The Parts of the Universe, and there they wrote down the number fourteen million, two hundred and thirty six thousand, seven hundred and thirteen.

Alice Peasgood

So that’s really the oldest record of a number that’s written in a modern way?

Hilary Holmes

That’s right.

Alice Peasgood

That sounds very advanced with what we have been talking about so far. I mean how did they use this number system?

Hilary Holmes

It enabled them to write down all sorts of different numbers, some very big numbers and that helped them in all sorts of calculations to do with measurement, the earth and so on.

Alice Peasgood

Once we have the digits nought to nine, and the place value system, somewhere along the line somebody must have put a zero in because if you haven’t got any hundreds you have got to put a zero in that column. So did that happen around about that time as well?

Hilary Holmes

That's an essential part of the place value system because if for example in a number you don't have any hundreds then you do have to put the zero there to ensure that the other numbers are in the correct columns.

Alice Peasgood

But the thing is you are talking about this being developed in India but we call the numbers nought to nine Arabic numerals. So what has happened there?

Hilary Holmes

As people realised how useful they were for calculations and so on, the numbers did spread across to the West and they reached Iraq and Baghdad in about 800 AD. There, there was a mathematician called Al-Khwarizmi who started to use these new numbers and actually wrote a book explaining how to add and subtract with the numbers. He went on further to develop all sorts of other mathematics like what we now know as algebra and that area became another centre of mathematical development in addition to the Indian centre.

Alice Peasgood

So what we are really saying is there are two major cultures which contributed hugely to the development of mathematical ideas because we are going way beyond numbers for counting here. I mean algebra is actually an Arabic word. So it has been carried through into our modern language. And the Islamic world actually has made a huge contribution to mathematical thought over the centuries.

Hilary Holmes

It was so important that Al-Khwarizmi's book was then translated into Latin and that’s how the use of the number system spread across to Europe.

Alice Peasgood

Now you have mentioned Latin I want to ask about Roman numerals because we see them on carvings and on walls all over the place. How did the Romans cope with MCXV when they did any counting?

Hilary Holmes

Well if you imagine trying to do any sort of calculations with Roman numerals it's very difficult. So they only used those numerals on documents and buildings and so on. When they actually had to do any calculating they would use a counting table. Nevertheless, the Roman numeral system did exist for thousands of years and it was what people felt familiar with. So when the new, Arabic system or Indian system arrived, there was some resistance to

using it. One of the most important developments there was actually by an Italian mathematician called Fibonacci and he also wrote a book that described how to use the Arabic number system which helped people to get to grips with this new form of calculating.

Alice Peasgood

And someone told me that was actually one of the most popular books in its time. I mean it could be seen as a mathematical best seller in its own terms…

Hilary Holmes

…Because it was so useful, because it enabled people who were trading to do the calculations quickly and efficiently.

Alice Peasgood

What intrigues me is that Roman numerals have still survived from monuments and if you look very carefully at the end of a television programme or a film they are still used for the copyright date at the end.

Hilary Holmes

Which just shows the dominance that they had and their resistance to change.

Alice Peasgood

So what we are saying here is that really the history of the development of number and mathematical ideas started out with counting and people wanting to do practical things and then as writing developed they found new ways of representing numbers and then started solving problems with them. So all of these cultures that were trading with each other over thousands of years have swapped ideas, found different ways of solving problems and that’s how the maths has developed. But is the story continuing? I mean what about more modern ideas like negative numbers?

Hilary Holmes

Well negative numbers first arose in trade and so on but when people first started to use them they found that it was a very difficult concept and actually referred to them as fictitious numbers because they were so difficult to understand and it wasn’t until the eighteen hundreds that negative numbers were really placed on a firm footing. So they really are quite a recent development.

Alice Peasgood

So that’s only two hundred years ago which is hardly any time at all compared with some of the dates we have been talking about. Is that the end of the story? Have we stopped doing more new numbers now or is there anything else?

Hilary Holmes

The story is carrying on all the time. I mean if you look back to the history, numbers have developed as people have needed to solve new problems and some of the developments that we have got now are things like the binary system because at the moment our society is dominated by computers and they use the binary system which just involves two digits—zero and one. So in a way we have come from a base sixty system to a base ten system and now we are heading towards a base two system. The other development is looking at numbers which are infinitely large or infinitesimally small and mathematicians this century have been researching into this kind of number which is known as the hyper-real numbers.

Alice Peasgood

So, we are going to continue to develop new numbers, as there are new problems to solve; that the story never really quite comes to an end.

Hilary Holmes

Well that’s the excitement of mathematics. As you have said when you have a new problem, you have got to find new ways of dealing with it. Who knows where it will lead in the future?

End transcript
 
Interactive feature not available in single page view (see it in standard view).
Hint symbol
Discussion

Keep track of the important places and dates. How did a particular civilization count? If you need to, feel free to pause the audio track to jot down notes.

Solution symbol
Answer

The main ideas, dates, and places mentioned in the audio track shown on the time line below.

The Rhind papyrus dates from just over 3500 years ago, and Fibonacci wrote his book Liber Abaci in 1202. Mathematical developments, including other place value systems, have also taken place in China and South America. The history of how mathematics has developed over the centuries is a long and fascinating one that involves many different cultures.

2.1.2 Ways of Representing Numbers

Activity: Fourteen Million

About 1500 years ago, in an Indian book on cosmology called The Parts of the Universe, the number fourteen million, two hundred thirty-six thousand, seven hundred thirteen was written down. How would you write this number today?

Hint symbol
Discussion

Okay, take this slowly. Be sure to change the words into numbers and that will help. Like this:

  • Fourteen is 14 and represents the ten and one millions.
  • Two hundred thirty-six is 236 and represents the hundred, ten, and one thousands.
  • Seven hundred thirteen is 713 and represents the hundreds, tens, and units (ones).
Solution symbol
Answer

We write numbers out like this:

Ten millionsMillions Hundred thousandsTen thousandsThousandsHundredsTensUnits
10,000,0001,000,000100,00010,0001,000100101
14236713

Thus, the number is written 14 236 713. Notice how leaving a small gap between each group of three digits makes the number easier to read. In the United States, we place commas between the groupings, like this: 14,236,713.

The table above is actually a place value table. You may have noticed that the value of each column is ten times the value of the column to its right. We’ll look at this in moment.

You can extend our number system indefinitely for larger whole numbers by adding extra columns on the left and labeling the columns as hundred millions, billions, ten billions, and so on.

2.1.3 Place Value

The value of a digit depends on its place, or position, in the number. Each place has a value of ten times the place to its right. Each position is referred to as a place holder.

You can see in the illustration above that the following is true:

  • 1 is in the ten thousands place.
  • 5 is in the thousands place.
  • 7 is in the hundreds place.
  • 6 is in the tens place.
  • 4 is in the units place.

…  and the number is 15,764, which is spoken as “fifteen thousand, seven hundred and sixty-four.”

So you can now see why zero is so important!  We need it to show that there is nothing at a certain place holder.  Without a zero how could we write 203 or even 10?

Activity symbolActivity: Writing Numbers

In your math notebook, write each number in two other ways.

(a) 92,400

Hint symbol
Discussion

You can use words! Each place holder has a name. Remember to count the places carefully moving from left to right.

Solution symbol
Answer
  • (a) 92,400.
    • 9 ten thousands, 2 thousands, 4 hundreds.
    • Ninety-two thousand, four hundred.
    • 90,000 + 2,000 + 400.

(b) One hundred fourteen thousand, six hundred sixty.

Solution symbol
Answer
  • (b) One hundred fourteen thousand, six hundred sixty.
    • 1 hundred thousand, 1 ten thousand, 4 thousands, 6 hundreds, and 6 tens.
    • 100,000 + 10,000 + 4,000 + 600 + 60.
    • 114,660.

(c) Three million, two hundred and four thousand and sixteen.

Solution symbol
Answer
  • (c) Three million, two hundred and four thousand and sixteen.
    • 3 millions, 2 hundred thousands, no ten thousands, 4 thousands, no hundreds, 1 ten and 6 units.
    • 3,000,000 + 200,000 + 4,000 + 10 + 6.
    • 3,204,016.

2.1.4 Numbers in Standard Form

In the United States, a number in standard form is separated into groups of three digits using commas. Each of these groups is called a period. So in the last section you will have seen that one hundred fourteen thousand, six hundred sixty was written as 114,660.

Here’s a number that is not written in standard form: 100210120. To re-write this in standard form we start from the right and group the numbers into sets of threes:

  • The first period is 120.
  • The second period is 210.
  • The third period is 100.

Thus, the number is 100,210,120.

Compare how both numbers are written. The version in standard form is much easier to understand, which is why we use this system to write numbers.

You may also find that instead of using commas to separate each period a space is left, for example 345 678 943.  This is the convention in some countries to avoid confusion where commas are used in place of decimal points in numbers.

Activity symbol Activity: Standard Form

In your math notebook, first identify the periods in the number 345678943 and then write it out in standard form.

Hint symbol
Discussion

Remember to group in sets of threes, starting from the right and moving from the left. Then separate each period using commas.

Solution symbol
Answer
  • The first period is 943.
  • The second period is 678.
  • The third period is 345.

Put them all together and you have 345,678,943.

Check out this game to see if you’ve conquered place values!

2.1.5 The Decimal System Compared

Activity symbolActivity: Different Countries, Different Methods

Try writing the numerical value for 1 billion using the number system with which you are familiar.

Hint symbol
Discussion

Your answer will depend on where you live. The system used in English-speaking countries is different from that used in other countries (like France, and Germany). In these other countries, a billion (“bi” meaning two) has twice as many zeros as a million; a trillion (“tri” meaning three) has three times as many zeros as a million, etc.

Note: The scientific community, in general, uses the American system.

Number of ZerosU.S. & Scientific CommunityOther Countries
3thousandthousand
6millionmillion
9billion1000 million (sometimes called 1 milliard)
12trillionbillion
15quadrillion1000 billion
18quintilliontrillion
Solution symbol
Answer

In English-speaking countries and the scientific community worldwide, one billion would be written “1,000,000,000.” In some other countries, one billion might be written “1,000,000,000,000.”

notepad symbol Activity: Using Your Math Notebook

Now is a good time to think about how you are using your math notebook. Are you using it to write down any unfamiliar math language? If you have a separate section in your notebook for this you will build up a list of math language—your own glossary of terms that you can look back to at any time if you need a reminder about the meaning of a word or phrase. Try this now with the section that you have just covered.

2.2 Using a Number Line

A number line can help you to visualize many different kinds of numbers. For example, on the number line below, the intervals between the whole numbers (or units) have each been split into ten equal intervals: These are tenths. If each tenth is then split into ten equal intervals, each of the smaller intervals will be hundredths, since there will be 100 of these intervals in a whole unit. The number line shows the numbers two-tenths, one unit and three-tenths, one unit and thirty-five hundredths, and one unit and eight-tenths.

To write these numbers in decimal form, the place value table can be extended by adding columns to the right, as shown below. Since the value of each column is ten times smaller than the value of the column to its left, the columns to the right of the units column will represent tenths, then hundredths, thousandths, and so on.

Let’s look at a specific example. Notice that the value of the digits is based on the number ten, even those to the right of the decimal.

You can see in the illustration above that the following is true:

  • 1 is in the tens place.
  • 7 is in the units place.
  • 5 is in the tenths place.
  • 9 is in the hundredths place.
  • 1 is in the thousandths place.

With decimals, different terminology can also be used to describe a particular placeholder. Often, you will need to round a decimal to a specific position, that is decimal place. Instead of asking you to round to the nearest tenth, you might see instructions to “round to 1 decimal place” (1 d.p.). Below is a table to help you keep track of the math vocabulary.

TenthHundredthThousandthTen Thousandth
1 d.p.2 d.p.3 d.p.4 d.p.

notepad symbolActivity: Decimal Numbers and Decimal Places

In your math notebook write the place values for each of these numbers. 

  • (a) 5.603.
Hint symbol

Discussion

Write the numbers in a place value table first starting from the units and working to the right.

  • (b) 10.986.
Solution symbol

Answer

(a) 5.603.

  • 5 is in the units place.
  • 6 is in the tenths place.
  • 0 is in the hundreths place.
  • 3 is in the thousandths place.

(b) 10.986.

  • 1 is in the tens place.
  • 0 is the units place.
  • 9 is the tenths place.
  • 8 is the hundreths place.
  • 6 is the thousandths place.

Now identify in parts (a) and (b) which number is in the second decimal place (2 d.p.).

Hint symbol

Discussion

Starting at one count each number to the right of the decimal point this will tell you the decimal place for each number.

Solution symbol

Answer

(a) 5.603.

0 is in the second decimal place.

(b) 10.986.

8 is in the second decimal place.

2.2.1 Converting Fractions to Decimals

Let’s relate fractions to decimals. If we have a whole pie and one divided by two a pie, we have 1 and one divided by two of a pie. Not only is that hard to read and type; it’s also hard to say! So, to represent whole numbers and fractions of whole numbers, we use decimals as we have just seen in the last section. We do this by using a decimal point to separate the whole number from the fraction.

For example, say you needed to write out 2 and three divided by 10 (three tenths) as a decimal. The whole part is 2 and the fractional part is three divided by 10. Thus, we would write the 2 to the left of the decimal point and the fractional part to the right of the decimal point. Thus, it would be written as 2.3.

Pencast symbolLet’s reinforce this concept with more explanation (Click on “View document”).

View document

Now try a few conversions on your own. Check out the next activity.

Activity symbol Activity: Fractions to Decimals

Rewrite each of the following fractions as a decimal.

(a) two divided by 10.

Hint symbol
Discussion

If the number does not have a whole number part, a zero is written in the units. This makes the number easier to read (it’s easy to overlook the decimal point). The first placeholder to the right of the decimal is the tenths. How many tenths do you have for each given number?

Solution symbol
Answer

(a)two divided by 10 equals 0.2 (because there are zero whole parts).

(b) three divided by 10.

Solution symbol
Answer

(b) three divided by 10 equals 0.3 (because there are zero whole parts).

(c) 1 and two divided by 10.

Solution symbol
Answer

(c) 1 and two divided by 10 = 1.2.

(d) 3 and 3/100.

Solution symbol
Answer

(d) 3 and 3/100 = 3.03.

2.2.2 Different Ways to Write Numbers

Check this out: We can write numbers in all kinds of ways!

Decimal NumberWritten as a FractionWritten in Terms of Place Value
4.1
4 units plus 1 tenth
12.04
1 ten plus 2 units plus 0 tenths plus 4 hundredths
0.001
one divided by 1000
1 thousandth

Activity symbol Activity: Writing Numbers in Different Ways

Write this table in your notebook and fill in the blanks following the example provided above.

Decimal NumberWritten as a FractionWritten in Terms of Place Value
5.1
18.05
0.004
Hint symbol
Discussion

Pay close attention to the position of each number and try writing in terms of place value first. You can also ask for help from friends and family. Be sure to explain place value to them.

Solution symbol
Answer
Decimal NumberWritten as a FractionWritten in Terms of Place Value
5.1
5 plus 1 tenth
18.05
1 ten plus 8 units plus 0 tenths plus 5 hundredths
0.004
four divided by 1000
4 thousandths

2.2.3 Expressing Numbers

Activity symbol Activity: Representing Numbers Part I

Using your math notebook, try the exercise below.

Solution symbol
Answer

In Unit 6, we will take a closer look at fractions and practice calculations with them.

2.2.4 More Practice with the Number Line

Let’s try a few more examples to make sure you are comfortable identifying the numbers.

Activity symbolActivity: Representing Numbers Part II

Sketch the number line provided below in your math notebook and try the following:

(a) What numbers are indicated by A, B, and C?

Hint symbol
Discussion

Take your time. Refer to the example on the previous screen and trace your steps. Identify the whole number to the left of B: this will be the whole part of the number. Since the unit is broken into ten equal parts, count the tick marks. This will represent the number of tenths, giving you the decimal portion.

Solution symbol
Answer

(a) A marks the number 0.7, B marks the number 0.15, and C marks the number 1.5.

(b) Mark the numbers 0.5, 1.2, and 1.85, on the number line.

Answer

(b) The numbers 0.5, 1.2, and 1.85 are indicated on the number line below.

2.2.5 Decimals in Daily Life

Think about when and how often you encounter decimals on a daily basis. You might meet decimals in your everyday life when you are shopping, such as prices or discounts, or measuring something, such as how tall you are or how far it is from your home to college. Try to think of a few and ask your friends what they use decimals for or where they have seen them.

Activity symbolActivity: Sunflowers and Winners

In a sunflower competition, Ahmed, Bert, and Cathy had the three tallest sunflowers. The heights of the sunflowers were measured; Ahmed’s is 1.8 feet tall, Bert’s is 1.67 feet tall, and Cathy’s is 1.72 feet tall. Which is the tallest sunflower? Which of these is the shortest?

Hint symbol
Discussion

Create a table that breaks down each height into its placeholders.

Solution symbol
Answer

To decide which of these three numbers is largest, you can either draw a number line and mark the heights on it, or use a place value table as shown below.

UnitsTenthsHundredths
Ahmed180
Bert167
Cathy122

Starting at the left-hand column, compare the digits in each column.

All the digits in the units column are 1, so you need to look at the tenths column. The first number has eight tenths, the second number has six tenths and the final number has seven tenths.

  • First place: Ahmed with the biggest sunflower at 1.8 feet.
  • Second place: Kathy with the second biggest sunflower at 1.72 feet.
  • Third place: Bert with the third biggest sunflower at 1.67 feet.

2.3 Rounding

The precision and reporting of measurements depends on the tools that are used and the purpose of the measurement. In this section, we will study the impact and the importance of rounding.

The world population is the total number of living humans on the Earth, currently estimated by the United States Census Bureau to be 6.95 billion as of July 1, 2011.That is not an exact value—people are being born (and dying) every second—but it does give you an indication of the world’s population.

In newspapers and elsewhere, numbers are often rounded so that people can get a rough idea of the size, without getting lost in details. You use rounding in your everyday life, such as rounding prices in a supermarket ($2.99 is about $3) or distances (18.2 miles is about 20 miles), or time (it took about 40 minutes to get there, as opposed to 38.2 minutes).

Using approximate values is useful if you want to get a rough idea of an answer before you get out scratch paper or a calculator. This estimation acts as a check on your calculation and may help you to catch any errors you may have made—maybe you pressed a wrong key or reversed numbers.

In this section, you are going to look at rounding, and in the next section we will use estimation to solve problems.

Reference

U.S. Census Bureau, “World POPClock Projection.” Available online at http://www.census.gov/ population/ popclockworld.html (accessed September 24, 2012).

2.3.1 How Far is It?

Suppose you are planning a journey by road from Washington, DC to Richmond, Virginia. According to the map, it’s approximately 112 miles. Depending on where you start in Washington DC and where you end in Richmond, the total distance could be more or less than 112 miles.

Assume we start from Location A in south-west Washington, DC and drive to downtown Richmond, Virginia. The actual mileage is 111.6. We would round up to 112 miles, because the number after the decimal point is greater than or equal to 5 and is therefore closer to 112 than 111. By the same token, if we traveled a little further to the south of the James River in Richmond and the mileage was 112.3, we would not round up, because the number after the decimal point is less than 5 and this time is closer to 112 than 113.

The general rules for rounding are:

  • Locate the place value to be rounded.
  • Look at the next digit to the right (the next smaller place value).
  • If that digit is a 5 or greater, “round up” the previous place value digit—which means to increase it by one.
  • If that digit is a 4 or less, leave the previous place value digit unchanged.
  • Replace all digits after the place value digit with 0 but only if these are to the left of the decimal point

Here’s a couple of examples to have a look at.

We want to round the world population figure of 6.95 billion to the nearest billion. So the place value that we want to round in this case is the units as these represent the billions, which is a 6. The next number to the right of this is a 9, which is 5 or greater, so we need to round up the 6 to 7.

So 6.95 billion rounded to the nearest billion is 7 billion.

Now we want to round 1.72 to the nearest tenth (or to 1 decimal place). This means rounding  the 7 in the tenths place. To the right of this is a 2, which is less than 5, so we leave the 7 as it is.

So 1.72 rounded to the nearest tenth (or to 1 decimal place) is 1.7

Notepad symbol Activity: Rounding

Round each of the following numbers to the specified placeholder.

(a) Round 126.43 to the nearest tenth.

Hint symbol
Discussion

Remember that the tenths place is located immediately to the right of the decimal point and is the same as rounding to one decimal place, or “1 d.p.” Be sure to use the digit in the hundredths place to make your final decision.

Solution symbol
Answer

(a)

  • Locate the place value to be rounded—in this case, the number 4.
  • Look at the next one to the right—in this case, it’s 3.
  • 3 is less than 5, so we round down, and the answer is 126.4.

(b) Round 0.015474 to the nearest thousandth.

Hint symbol
Discussion

You will use the placeholder immediately to the right of the thousandths place to make your rounding decision. This is the same as rounding to 3 d.p.

Solution symbol
Answer

(b)

  • Locate the place value to be rounded—in this case, the number 5.
  • Look at the next one to the right—in this case, it’s 4.
  • Because 4 is less than 5, we round down, and the answer is 0.015.

(c) Round 1.5673 to 2 decimal places (2 d.p.).

Solution symbol
Answer

(c)

  • Locate the decimal place to the rounded—in this case the number is 6.
  • Look at the next one to the right—in this case, it’s 7.
  • Because 7 is greater than 5, we round up, and the answer is 1.57.

(d) The speed of light is 299,792.458 km/sec. Round that number to the nearest 1000 km/sec.

Hint symbol
Discussion

You will need to round to the nearest thousand (not thousandth). Be careful!

Solution symbol
Answer

(d)

  • Locate the place value to be rounded—it’s the second 9.
  • Look at the next one to the right—it’s 7.
  • Since 7 is more than 5, we round up. But rounding up 9 makes it 10, so we have to round up the next 9, which makes that 10 as well so we have to round up the 2 to 3.
  • So the answer is 300,000 km/sec. Notice that we write these new zeros, as they are to the left of the decimal point.

Videoclip symbolFor more exposure and practice on rounding, check out Professor Perez and see if you can get the hang of rounding and following these rules! Feel free to skip ahead in the video if you feel comfortable the material.

Interactive feature not available in single page view (see it in standard view).

2.3.2 Calculator Exploration: Decimals and Rounding

In the following exploration, you will learn how to work with decimals on the calculator, and you will revise rounding your answers.

Calculator symbol The calculator can be accessed in the left-hand side bar under Toolkit.

Calculations involving decimals are often more complicated than calculations using just whole numbers, and this is when the calculator is very useful. You enter the decimal point using the button marked Period symbol . You’ll find this below the numbers on the calculator.

On your keyboard, use the period key.

Mathematical OperationCalculator ButtonKeyboard Key
Decimal point
Period key
Period key

Now it’s time to try some calculations. Start with an easy one which you can do in your head:

1.3 plus 2.5

The key sequence is:

Enter this on the calculator and check that you get the answer 3.8.

Activity symbol Activity: Working with Decimals

Use the calculator for these calculations.

(a) 21.74 plus 16.6 minus 19.27

Hint symbol
Discussion

Remember to clear each calculation before you start the next, using the Cancel symbol button.

Solution symbol
Answer

(a)21.74 plus 16.6 minus 19.27 equals 19.07

(b) 31.97 multiplication 0.264

Solution symbol
Answer

(b) 31.97 multiplication 0.264 equals 8.44008

(c) 21.9 division 6.7

Solution symbol
Answer

(c) 21.9 division 6.7 equals 3.26865671641791to 14 decimal places!

In the solution to part (c), you will have noticed that the calculator gave the answer in fraction format before converting it to a decimal. Don’t worry about this at the moment; it will be explained in Unit 7.

The calculator also displayed 14 decimal places in the answer. There are more digits after this, but 14 digits are more than enough for most purposes! In fact, 14 are usually too many, and so we round the answer to give a shorter, more useful form.

For example, if we were to round the displayed answer to 2 d.p., then we would look at the third decimal place, which is 8, to make our decision. Because 8 is larger than 5, it indicates that we would need to round up, giving us 3.27 as a rounded answer.

2.3.3 More Practice Rounding

Activity symbolActivity: Rounding Your Answer

Round the answer to the last calculation 3.26865671641791 to:

(a) The thousandths place holder.

Hint symbol
Discussion

Find the digit in the thousandths place holder. Then look at the first digit that you will cut off. If it is 0, 1, 2, 3, or 4, then leave the number as it is; if it is 5, 6, 7, 8, or 9, then round up the final digit of your answer by 1. (You can ignore all the later digits that you are cutting off, because they are to the right of the decimal point.)

Solution symbol
Answer

(a) Count 3 decimal places (this is the thousandths place) and look at the next digit. 

The first digit to be cut off is 6, which falls in to the “5 or more” category, so you will round up the final digit of your answer, meaning you will increase it by 1. The answer, rounded to 3 d.p., is 3.269.

(b) 2 decimal points.

Solution symbol
Answer

(b) Count 2 decimal places (this is the hundredths) and look at the next digit.

Image showing 3.26865671641791 and first digit to be cut so 3.268

The first digit to be cut off is 8, (which is again “5 or more”), so round up the final digit of your answer by adding 1. The answer, rounded to 2 d.p., is 3.27.

Here are a couple more examples to give you practice at rounding.

Activity symbol Activity: More Rounding

Carry out the following calculations and write down the full calculator answer. Round the answer to 1 d.p. first. Then, round the calculator answer to 2 d.p.

(a) 2.1895 division 0.37

Hint symbol
Discussion

There are lots of numbers here, so take your time entering them into the calculator. Watch the screen to make sure that you are entering them correctly, and correct any mistakes as you go along.

Solution symbol
Answer

(a) 2.1895 division 0.37 equals 5.91756756756757

Calculator screen showing 2.1895 over 0.37 = 5.91756756756757

To round the answer to 1 d.p., look at the first digit that you will cut off.

It is 1, so leave the previous digit as it is. The answer, rounded to 1 d.p., is 5.9.

To round the answer to 2 d.p., look at the first digit that you will cut off.

It is 7, so round the last digit of the answer up by adding 1. The answer, rounded to 2 d.p., is 5.92.

(b) 64.935 multiplication 230.6 division 0.077

Solution symbol
Answer

(b) 64.935 multiplication 230.6 division 0.077 equals 194467.67532467532468

Notice that the full calculation does not fit in the black window, so the final digits are missing. However, the full answer appears below in the white window.

To round the answer to 1 d.p., look at the first digit that you will cut off.

It is 7, so round the previous digit of the answer up by adding 1. The answer, rounded to 1 d.p., is 194467.7.

To round the answer to 2 d.p., look at the first digit that you will cut off.

It is 5, so we round up. The answer, rounded to 2 d.p., is 194467.68.

Activity symbol Activity: Rounding Brainstretcher—An Interesting Observation!

You may have noticed that the full answer to part (a) in the previous activity has a repeating pattern of the digits 756. Why does the final digit break this pattern? Why do you think it is 7, not 6?

Hint symbol
Discussion

Think about rounding to 14 d.p.

Solution symbol
Answer

When the calculator rounds the answer to 14 d.p., the first digit that it cuts off (following the pattern) is 7. This is “5 or more,” so the final digit in the answer is rounded up from 6 to 7.

To recap:

  • To enter a decimal, use the Period symbol button on the calculator or the period key on your keyboard.

The calculator will give answers to 14 decimal places. This is usually much too precise, so round your answer, either as the question asks, or, for a practical calculation, to a sensible number of figures. For example, if working with money, rounding the nearest (cent) hundredth is reasonable.

2.3.4 Rounding in the Real World

Activity symbol Activity: How Much is It?

Suppose you want to buy a new truck. When you visit the dealership’s website, it says the cost of the vehicle you are looking at is $29,748. When you call the dealership on Thursday, the sales representative, Joe, tells you the cost is $29,750 and the next day another sales representative, Tina, informs you that the car costs $29,700. Why did the quote you were given on the cost of the truck change?

Hint symbol
Discussion

Round the website cost to the closest $10. Then round the website cost to the closest $100.

Solution symbol
Answer

Here’s how the variation in the quotes happened:

Rounding to the closest $10 (this is the same as rounding to the tens): The digit 4 is in the tens place. Looking to the next smaller place value (to the right), the digit is an 8. Since 8 is larger than 5, we round up the 4 to 5 in the tens place. Finally, we replace the units digit with 0 (because this place is to the left of the decimal point). The rounded number is $29,750. This explains Joe’s quote.

Rounding to the closest $100 (this is the same as rounding to the hundreds): The digit 7 is in the hundreds place. Looking to the tens place (to the right of the 7), the digit is a 4. Since 4 is less than 5, we leave the digit 7 unchanged and replace the digits after the 7 with zeros. The rounded number is $29,700. This explains Tina’s quote (which is more appealing than Joe’s since it’s less).

Want to be a mathionaire? Check out this rounding game for a little extra practice!

Notepad symbol Activity:  Thinking About Your Learning

It is a good time now you are nearly at the end of the Unit 2 to think about how your learning has gone so far.  Maybe you think that you need to change some aspects of how you are studying—is your math notebook organised in a way that is helping you?  Have you found the best time to study for you?  Do you think that you might need to take more notes?

Jot down some thoughts now in your math notebook that you can look back on when you start Unit 3.  

So far, we’ve studied how to write and estimate numbers. Now we will begin to perform calculations with them. First, let’s look at addition and subtraction of numbers.

2.4 Self-Check on Unit 2

The more you practice, the more your skills improve. Below are some exercises that will help you continue to develop your ability and check to make sure you understand the concepts discussed in this unit. Be sure to write your work out in your math notebook so that you can refer to it later if necessary.

2.4.1 Self-Check on Using a Number Line

Tick symbolExercise 1: Reading from a Number Line

Look at this number line and write down what three numbers are shown.

Solution symbol
Answer

(a) 0.25

(b) 1.95

(c) 0.9

Tick symbolExercise 2: Plotting on a Number Line

Sketch the following number line in your math notebook.

Mark the following numbers:

(a) 0.45

(b) 1.2

(c) 0.825

Solution symbol
Answer

2.4.2 Self-Check on Rounding

Tick symbolExercise 3: Rounding Decimal Numbers

Round each number to the specified placeholder:

(a) 456.28377 to the nearest thousandth.

Solution symbol

Answer

(a) 456.284 (7 in the ten thousandths place indicates we round up).

(b) 2885 to the nearest hundred.

Solution symbol
Answer

(b) 2900 (8 in the tens place indicates we round up).

(c) 43.653 to the nearest unit.

Solution symbol
Answer

(c) 44 (6 in the tenths place indicates we round up).

(d) 752.245 to the nearest tenth.

Solution symbol

Answer

(d) 752.2 (the 4 in the hundredths place indicates we leave the digit in the tenths place unchanged).

(e) 10.995 to 1 decimal place.

Solution symbol

Answer

(e) 11.0 (the 9 in the second decimal place indicates that we round up 9 to 10).

(f) 15.5435 to 3 decimal places.

Solution symbol

Answer

(f) 15.544 (the 5 in the fourth decimal place indicates that we up to 3 to 4).

(g) There is an advert in your local newspaper for a used car. The advertised price is $10,649.  Estimate the price to the nearest $100 and $1000.

Solution symbol

Answer

$10,600 (to the nearest $100).

$11,000 (to the nearest $1000).

2.5 Quiz Time

Practice Quiz

Now that you have taken the time to work through these sections, do this short quiz. It will help you to monitor your progress, particularly if you took the quiz at the start of the unit as well.

The quiz does not check all the topics in the unit, but it should give you some idea of the areas you may need to spend more time on. Remember, it doesn’t matter if you get some, or even all of the questions wrong—it just indicates how much time you may need to come back and review this unit!

Click here for the post quiz.

2.6 Study Checklist

Study Checklist

Read through the list below and think over all the work you have done in this unit. If there is a checkpoint that doesn’t seem familiar, skim your notes to jog your memory. Remember that your mathematical skills will develop and grow stronger over time. Just keep working at it!

You should feel confident to:

  • Read and write decimal numbers.
  • Understand and use a number line.
  • Use rounding and estimation.
  • Start to use the calculator.

Good start! You’re doing a great job.

Outlook on Unit 3

In the next unit we will move on to look at calculations using the four basic operations of addition, subtraction, multiplication, and division. See you there! Click here to begin Unit 3.